Figure 1.
Schematic of variations in muscle activity and limb forces with altered stance distance during balance tasks in cats hypothesized to arise from neuromechanical interactions.
Top to bottom: sagittal-plane kinematics, left hindlimb ground reaction forces, left hindlimb muscle tuning curves. As stance distance between the fore- and hind-limbs is decreased from left to right (top row), a wider range of ground reaction force directions is observed (middle row), as well as increased muscle activation; however, muscle tuning to perturbation direction is conserved (bottom row).
Table 1.
Hypothesized models of optimal task-level control.
Figure 2.
Experimental postural perturbation paradigm and example data used for model constraints and validation.
A: Directions of translational perturbations are evenly-spaced in the horizontal plane. B: Coordinate system for forces and kinematics. C: Time traces of platform position, CoM and CoP displacement for a 60° perturbation along the direction of the perturbation, and left hindlimb ground reaction forces for 20 perturbations (cat bi) in the preferred postural configuration. The shaded region represents the initial period of active force generation due to the postural response. The CoM and CoP values in the time bin shown were used to define constraints on performance of the quadrupedal model, and individual forces across the four limbs were then compared to model predictions.
Table 2.
Summary of muscles included in the hindlimb model and analyzed in experimental data.
Figure 3.
Kinematic and kinetic constraints used in the optimal control models.
A: Kinematics of the musculoskeletal model parameterized to cat bi at four stance distances. LH, left hindlimb; LF, left forelimb; RF, right forelimb; RH, right hindlimb. B: average forces and moments at the CoM in each perturbation direction. Solid lines indicate experimental data, dashed lines indicate task-level constraints used in models MMe, MMm, SMe, SMc. C: average displacement of the CoP in each perturbation direction. Solid lines indicate experimental data, dashed lines indicate task-level constraints used in model MPe.
Figure 4.
Limb forces predicted by optimal task-level control of CoM force and moment.
A: average horizontal plane forces observed in each postural configuration of cat bi (black) compared with model MMe predictions (green). Force vectors are drawn for each limb (clockwise from top left: LF, left forelimb; RF, right forelimb; RH, right hindlimb; LH, left hindlimb) with their origins offset in the direction of platform motion. Stance distance decreases from left to right. Predicted forces were directed towards and away from the CoM, characteristic of the force constraint strategy described previously [18] at longer stance distances (34 and 27 cm), whereas a wider range of force directions was observed at shorter stance distances (13 cm) [53]. B: comparison of average and predicted limb force components in polar coordinates. HF, horizontal force; VF, vertical force. Predicted horizontal plane forces reproduced the region of invariant force directions for perturbation directions that unloaded the hindlimb (180° to 270°) observed at longer stance distances (34 and 27 cm) (arrows).
Figure 5.
Predicted horizontal plane forces obtained by altering the cost function and the task level variable.
A: model MMm predictions (yellow). Note that unlike MMe predictions (Figure 4A), MMm predictions are approximately constant as stance distance decreases. Compare changes between 34 cm and 13 cm to changes in Figure 4A. B: model MPe predictions (blue). Note that predicted forces are near the anterior-posterior axis, the strongest axis of force production in the isolated hindlimb [20] in all perturbation directions and postural configurations.
Table 3.
Summary of average limb force R2 values predicted by each model formulation.
Figure 6.
Predicted horizontal plane forces obtained by controlling experimentally-derived muscle synergies versus individual muscles.
Top to bottom: average horizontal-plane forces observed in each postural configuration of cat ni (black), predictions of models controlling individual muscles: MMe (green), or muscle synergies: SMc (blue), and SMe (purple). Arrows highlight significant force magnitude reductions observed in data, SMc, and SMe, but not in MMe.
Figure 7.
Comparison of limb forces predicted by controlling experimentally-derived muscle synergies rather than individual muscles in polar coordinates.
Data correspond to horizontal-plane forces shown in Figure 6. Colors as in Figure 6. Note that LH HF magnitudes in perturbation directions that unloaded the left hindlimb (horizontal bars, 180° to 270°) exhibited a monotonic decrease in models SMc and SMe from the preferred (29 cm, solid lines) to the shortest (18 cm, shortest dashed lines) stance distance similar to data that was not predicted in MMe.
Figure 8.
Examples of left hindlimb muscle tuning to perturbation direction observed in data and predicted by optimal task-level control using individual muscles or experimentally-derived muscle synergies.
Top to bottom: experimental data, predictions of models MMm, SMe, and SMc. Colors as in Figure 6. All models predicted smooth cosine muscle tuning to perturbation direction similar to experimental data, particularly in morphologically simple extensors (e.g., VM, SOL) and in some flexors (e.g., PSOAS). However, some flexors were recruited only when muscle synergies were controlled rather than individual muscles. Compare BFP, GRAC, TA in MMe vs. SMe. Some multifunctional muscles were more difficult to predict; e.g., unlike experimental results [15], MG was recruited with pattern similar to a flexor muscle in all models, with ankle extension being provided by extensor-tuned SOL. Biarticular muscle SART is listed as a hip flexor because it is implemented as such in the musculoskeletal model.
Figure 9.
Observed and predicted changes in muscle tuning curve magnitude and direction across postural configurations.
A: Comparison of muscle tuning curve magnitude scaling across postural configurations observed in cat bi with scaling predicted by models MMm, SMe, and SMc. Data points for individual muscles are shown as filled circles. B: Comparison of muscle tuning curve peak direction shift across postural configurations observed in cat bi with direction shifts predicted in models MMm, SMe, and SMc. Note that although SMc predicted increased tuning curve shifting compared to MMe, none of the models predicted significantly increased shifting compared to experimental data. ns, p>0.05; *, P<0.05; ANOVA, post hoc tests.
Figure 10.
Comparison of predicted fits to limb force data, computation time and control effort required for task-level optimal control formulations.
Controlling experimentally-derived muscle synergies predicts similar force outputs, but with reduced computation time and increased control effort compared to controlling individual muscles. A: comparison of fits to limb force data predicted by models MMe, SMe, and SMc. LH, left hindlimb; RF, right forelimb. Colors as in Figure 6. B: comparison of average computation time and average sum-squared left hindlimb muscle activation predicted by models MMe, SMe, and SMc. Muscle activation values for each cat are normalized to 100% of the amount predicted by model MMe in the preferred postural configuration.